There will be more sufficeint guides online how to check these with pretty diagrams, but a simple check for both would be: -
Tyres - Each tyre will have a legal tread wear indicator on them and these can be found in the big grooves running round the circumference of the tyre. If your tread blocks are higher than these little raised indicators, then your tread is legal. Other things to look out for are badly cracked/blistered side-walls of your tyres.
Brakes - Pads, will require a little torch to check 'thickness'. If you can see pad between your brake disc and the pad backing, then you're ok for now. But if it looks under 2cm's then there is a chance the inside pad could be further worn. As for discs, you will be looking for a lip on the inside or outer edge of the disc...or grooving on either face.
Drums are a bit trickier, but I can guide you through that if you need to.
HTH, Ian